Abstract
The analysis of binary response data commonly uses models linear in the logistic transform of probabilities. This paper considers some of the advantages and disadvantages of simple least-squares estimates based on a linear representation of the probabilities themselves, this in particular sometimes allowing a more direct empirical interpretation of underlying parameters. A sociological study is used in illustration.
Highlights
The interpretation of data in the form of binary outcomes arises in many areas of science from the primary physical and biological sciences and their application through to more directly applied areas and the social sciences
Suppose that for n independent individuals, we observe a realization of a binary outcome variable Yi (1 i n) taking values 1 or 21, and that for individual i there is a p  1 vector xi of explanatory variables
A widely used representation is the linear logistic form in which logfpr(Yi 1⁄4 1)/pr(Yi 1⁄4 21)g is assumed to depend linearly on xi. This leads to a simple interpretation of regression coefficients as ratios of effects when the binary responses are concentrated at one of the two levels but otherwise the interpretation is less direct
Summary
The interpretation of data in the form of binary outcomes arises in many areas of science from the primary physical and biological sciences and their application through to more directly applied areas and the social sciences. A widely used representation is the linear logistic form in which logfpr(Yi 1⁄4 1)/pr(Yi 1⁄4 21)g is assumed to depend linearly on xi This leads to a simple interpretation of regression coefficients as ratios of effects when the binary responses are concentrated at one of the two levels but otherwise the interpretation is less direct. There are implicit restrictions on the parameter space, namely that for all data x, jbTxj 1 If both the linear in probability and linear logistic models give adequate fit, the former has the advantage that the linear regression coefficients have a clearer operational interpretation in terms of numbers of individuals potentially influenced by a unit change of an explanatory variable. For a further discussion concerning a similar model for Poisson variables, see [5]
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